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Langton's Ant

Jacek Strzelczyk. Langton's Ant. 6 th of November 2007. Algorithms, Logic and Complexity. Computer-generated mysteries. integers with interesting but unexplained divisibility properties observed geometric configurations with no proof of existence. 07-11-6. Jacek Strzelczyk.

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Langton's Ant

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  1. Jacek Strzelczyk Langton's Ant 6th of November 2007 Algorithms, Logic and Complexity

  2. Computer-generated mysteries integers with interesting but unexplained divisibility properties observed geometric configurations with no proof of existence 07-11-6 Jacek Strzelczyk

  3. The story so far ant – two-dimensional Turing machine lives on plane partitioned into squares (cells)‏ each cell can be in one of several states 07-11-6 Jacek Strzelczyk

  4. Ant's algorithm Change the color of the square you are standing on- (if the square is black change it to white and if it is white change it to black)‏ Now Walk Forward to the square in front of you Look at the color of the square you are standing on. If it is black turn Left and if it is white turn Right by 90 degrees. Return to step 1. http://www.tiac.net/~sw/LangtonsAnt/LangtonsAnt.html 07-11-6 Jacek Strzelczyk

  5. First phenomenon: patterns • centrally symmetric „track” The universe of ant at times 184, 368 and 472 07-11-6 Jacek Strzelczyk

  6. Second phenomenon: highway • highway to infinity 07-11-6 Jacek Strzelczyk

  7. n-states ants • n different states for cells to be in • rule-string programming • ant description e.g. LLRRRLR • if L = 1 and R = 0 then ant's „genome” LLRRRLR = 98 07-11-6 Jacek Strzelczyk

  8. Phenomenon described by Propp • surprising ants 9 and 12 [LRRL and LLRR] • patterns on visiting starting cell The universe of ant 12 at time 16,464 The universe of ant 9 at time 38,836 07-11-6 Jacek Strzelczyk

  9. More ants of this sort... • rule strings of length 6 • mystery: the rule strings that lead to bilaterally symmetric • patterns patterns are 33, 39, 48, 51 and 60 • all divided by three ! 07-11-6 Jacek Strzelczyk

  10. Proof idea: Truchet tiles • cells split into • * H-cells (entered horizontally, exited vertically)‏ • * V-cells (entered vertically, exited horizontally)‏ • Truchet tiles – schematic „switches” The initial state of universe 07-11-6 Jacek Strzelczyk

  11. Truchet contours • closed curves – Truchet contours 07-11-6 Jacek Strzelczyk

  12. Principal contour • a contour through starting point 07-11-6 Jacek Strzelczyk

  13. The Even Run-Length Property • Why do some ant tracks exhibit recurrent bilateral symmetry and others not? • Rule-strings for 4-state and 6-state ants that exhibit recurrent symmetry: • Ant Rule-string • 9 LRRL • 12 LLRR • 33 LRRRRL • 39 LRRLLL • 48 LLRRRR • 51 LLRRLL • 57 LLLRRL • 60 LLLLRR 07-11-6 Jacek Strzelczyk

  14. Diagonals graph • Two types of cells: • - cold – if its state is odd (won't change operation after the next visit)‏ • - hot – if its state is even (will change operation after the next visit)‏ • For hot cells display not only the Truchet tile but also its diagonal. 07-11-6 Jacek Strzelczyk

  15. Even diagonal-degree property • All vertices in the diagonals graph have even degree (0, 2 or 4)‏ • If the state of the universe satisfies the even diagonals-degree property • with the ant at home, then the ant must travel along the principal contour, • but when it completes this path and returns home, it restores the even • diagonals-degree property, so that it must once again travel along • the (new) principal contour, and so on, ad infinitum. 07-11-6 Jacek Strzelczyk

  16. References • „Further Travels with My Ant”, David Gale, Jim Propp, Scott Sutherland • and Serge Troubetzkoy, published in „Math. Intelligencer” 17 (1995)‏ 07-11-6 Jacek Strzelczyk

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